Testing the W -exchange mechanism with two-body baryonic B decays

The role of W -exchange diagrams in baryonic B decays is poorly understood, and often taken as insignificant and neglected. We show that charmful two-body baryonic B→BcB¯′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ B\to {\mathbf{B}}_c\overline{\mathbf{B}}^{\prime } $$\end{document} decays provide a good test-bed for the study of the W -exchange topology, whose contribution is found to be non-negligible; here Bc is an anti-triplet or a sextet charmed baryon, and B¯′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{\mathbf{B}}^{\prime } $$\end{document} an octet charmless (anti-)baryon. In particular, we calculate that ℬB¯0→Σc+p¯=2.9−0.9+0.8×10−6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{\mathcal{B}}\left({\overline{B}}^0\to {\Sigma}_c^{+}\overline{p}\right)=\left({2.9}_{-0.9}^{+0.8}\right)\times {10}^{-6} $$\end{document} in good agreement with the experimental upper bound. Its cousin B¯s0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\overline{B}}_s^0 $$\end{document} mode, B¯s0→Λc+p¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\overline{B}}_s^0\to {\Lambda}_c^{+}\overline{p} $$\end{document}, is a purely W -exchange decay, hence is naturally suited for the study of the role of the W -exchange topology. We predict ℬB¯s0→Λc+p¯=0.8±0.3×10−6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{\mathcal{B}}\left({\overline{B}}_s^0\to {\Lambda}_c^{+}\overline{p}\right)=\left(0.8\pm 0.3\right)\times {10}^{-6} $$\end{document}, a relatively large branching ratio to be tested with a future measurement by the LHCb collaboration. Other predictions, such as ℬB¯0→Ξc+Σ¯−=1.1±0.4×10−5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{\mathcal{B}}\left({\overline{B}}^0\to {\Xi}_c^{+}{\overline{\Sigma}}^{-}\right)=\left(1.1\pm 0.4\right)\times {10}^{-5} $$\end{document}, can be tested with future Belle II measurements.


Introduction
Decays of B mesons to multi-body baryonic final states, such as B → BB M and BB M M , where B (M ) represents a baryon (meson), have been richly studied. Their branching ratios are typically at the level of 10 −6 [1]. These relatively large branching ratios are due to the fact that the baryon-pair production tends to occur in the threshold region of m BB m B + mB , where the threshold effect with a sharply raising peak can enhance the branching ratio [2][3][4][5]. On the other hand, without the recoiled meson(s) to carry away the large energy release, the two-body B → BB decays proceed at the m B scale, several GeV away from the threshold region, resulting in the suppression of their decay rates [6,7]. So far, only three charmless modes have been seen experimentally:B 0 → pp (B O(10 −8 )) and B − → Λ ( * )p (B O(10 −7 )), with Λ * ≡ Λ(1520) [8][9][10].
The tree-level dominated B → BB decays can proceed through the W -exchange, emission and annihilation diagrams, depicted in figures 1(a,b,c), respectively. However, the W -exchange (annihilation) process is regarded as helicity suppressed [11,12], and hence neglected in theoretical studies [13][14][15][16][17][18]. Moreover, one also neglects the penguinlevel gluon-exchange (annihilation) contributions, which leads to B(B 0 s → pp) 0 [17]. Consequently, the observation of theB 0 s → pp decay would provide valuable information on whether contributions from the exchange (annihilation) processes play a significant role. The smallness of the current upper bound on its branching ratio, B(B 0 s → pp) < 1.5×10 −8 [8], indicates an experimentally difficult decay mode to study the role of exchange (annihilation) diagrams.
On the other hand, experimental data show that B(B → B cB ) (10 2 − 10 3 )B(B → BB ), where B c denotes a charmed baryon [1]. With significantly larger decay rates, charmful two-body baryonic B decays offer an interesting and suitable environment in which to study and test the role of the W -exchange (annihilation) mechanism. The set of measured B → B cB branching ratios is nevertheless scarce [1,19,20]: where (a,c) depict the W -exchange and annihilation processes, respectively, and (b) depicts the W -emission process, with q = (u, d, s) for As in the case of the charmless final states considered above, the B → B cB decays can proceed through both the W -exchange and W -emission diagrams. Again, theoretical studies regard the W -exchange diagram as helicity-suppressed, which is in analogy with the leptonic B → ν decays, and take the W -emission diagram as the dominant contribution [13,14,18,21,22]. For clarification, we present the amplitudes of B → ν and B → B cB with W -exchange contribution in figure 1a as where A(B → B cB ) by equation of motion is presented as a reduced form with the quark mass m c [23]. In eq. (1.2), the small m , with = (e, µ), is responsible for the helicity suppression in B → ν . Nonetheless, m c ∼ 1.3 GeV [1] in A(B → B cB ) is clearly helicity allowed, indicating that neglecting its contribution may not be a valid assumption to make. For completeness, we note that the theoretical studies in refs. [23][24][25][26] also considered the exchange and annihilation contributions in B → BB and D + s → pn. We therefore propose the study of the family of charmful two-body baryonic B → B cB decays to improve our knowledge of the role of W -exchange diagrams in B decays to baryonic final states. Since modes such asB 0 → Ξ + cΣ − andB 0 s → Λ + cp can only proceed via the W -exchange diagram, measurements of their branching ratios are direct tests of the W -exchange mechanism. JHEP04(2020)035

Formalism
Besides the W -emission (A em ) amplitudes studied elsewhere [18,21,22], we consider the often neglected W -exchange (A ex ) amplitudes for the charmful two-body baryonicB 0 (s) → B cB decays, where B c denotes the anti-triplet and the sextet charmed baryon states, (Ξ +,0 c , Λ + c ) and (Σ ++,+,0 , Ω 0 c ), respectively, andB an octet charmless (anti-)baryon. The decays with the decuplet charmless (anti-)baryons are excluded from the calculations in this paper due to the lack of the corresponding timelike baryon form factors.
We show in table 1 the amplitudes involved in the interestedB 0 (s) → B cB modes. The decay rate of modes that can only occur through the W -exchange diagram would be vanishingly small by construction if the importance of these diagrams was to be insignificant: None of these relations has yet been verified experimentally. The relevant part of the Hamiltonian for theB 0 (s) → B cB decays has the following form [27]: where G F is the Fermi constant, V ij stand for the CKM matrix elements, and (q 1 In the factorization approach, the W -exchange amplitude ofB 0 (s) → B cB JHEP04(2020)035 is given by [21,23] where q = d(s) forB 0 (s) , and a 2 = c eff 2 + c eff 1 /N c consists of the effective Wilson coefficients (c eff 1 , c eff 2 ) = (1.168, −0.365) and the color number N c . The matrix elements in eq. (2.3) are defined as [1,22] where f B is the B meson decay constant, q µ = (p Bc + pB ) µ the momentum transfer, and f i and g i (i = 1, 2, 3) are the timelike baryonic (0 → B cB ) form factors. The decay amplitude of B → B cB in the general form is written as with A S,P standing for the (S, P )-wave amplitudes. By only receiving the W -exchange contributions in eq. (2.3), the A S,P are given by where C S,P = ∓iG F a 2 V cb V * uq f B / √ 2, m ± = m Bc ± mB , and (f 2 , g 2 ) vanish due to the contraction of σ µν q µ q ν = 0. We use the timelike baryonic form factors with the light-front quark model [28], which have been widely applied to the b and c-hadron decays [29][30][31][32][33][34][35]. In the light-front frame, since the momentum transfer q µ is presented as q + = q 0 + q 3 = 0, it simply indicates the non-contributions from (f 3 , g 3 ) as in the case of the B → M transition [34,35]. This is in accordance with lattice QCD calculations, where the derivations of (f 3 , g 3 ) ∝ (f 1 , g 1 )/t show the suppression in the spacelike region [36], even though the corrections beyond leading order have been considered. Therefore, (f 3 , g 3 ) are often neglected in the literature [22,37].
W -exchange contribution to B(B 0 → Λ + cp ) should be at the level of 10 −5 . With N c = 2.9 (a 2 = 0.04), we are able to get B(B 0 → Λ + cp ) = (1.0 +0.4 −0.3 ) × 10 −5 with the errors from the uncertainties of the form factors. In the generalized factorization, one allows N c to shift from 2 to ∞ as an effective number to account for the non-factorizable strong interaction. Particularly, N c 3 implies a mild correction [38]. Accordingly, we estimate the previously neglected W -exchange contributions to B(B → B cB ), given in table 3.

Discussions and conclusions
Several suppression factors have been proposed to support the neglecting of the W -exchange (annihilation) contributions. The first one is in analogy with the semileptonic B → ν decays [12]. In fact, we obtain While m 2 µ is responsible for the helicity suppressed B(B − → µ −ν µ ) 10 −7 [39], B → B cB with m 2 + = (m Bc + mB ) 2 obviously allows for helicity-flip. The second one is the decay constant f B = 0.19 GeV in eq. (4.1), regarded as small and suppressing the W -exchange contribution [40]. However, it is found that B(B − → τ −ν τ ) with f B can still be as large as (1.09 ± 0.24) × 10 −4 [1]. Besides, the ratio of comes from (f 1 , g 1 ) ∝ (α s /t) 2 with t = m 2 B , which corresponds to the hard gluons that transfer the energy of m B [23,41]. In the B → B cB decays, however, (f 1 , g 1 ) with a charmed baryon are not small at t = m 2 B , as shown in eqs. (2.7) and (3.2). Note that (f 1 , g 1 ) in the timelike and spacelike regions can be associated with the crossing symmetry, and (f 1 , g 1 ) with the light-front quark model agree with those in lattice QCD calculations [28,36]. By contrast, B(B 0 → Λ + cp ) was once calculated as small as 4.6 × 10 −7 [42], taken as the another theoretical support for the neglect of the W -exchange diagram [12,22]. However, the estimation was done with the baryonic form factors in the dipole form of F (0)/(1 − t/m 2 D * ) 2 adopted from the Λ + c → p transition [42], such that the momentum transfer at m B exceeds the D * meson pole, causing the suppression, whereas the validity of the D * meson pole has never been tested in the timelike region. Besides, F (0) was not clearly given, due to the lack of the studies on the quark and QCD models at that time.
With the W -exchange contributions, we obtain where B(B 0 → Σ + cp ) is consistent with the current data in eq. (1.1). In particular, B(B 0 → Σ 0 cn ) is predicted to be ten times larger than the pole model calculation [22]. On the other hand, by only considering the W -emission contribution, the pQCD approach gives that B(B 0 → Λ + cp ) = (2.3 − 5.1) × 10 −5 [18], which is more than 4 standard deviations away from the measured central value. It is interesting to know if the pQCD approach would overestimate B(B − → Σ 0 cp ) as well. One way of comparing the W -emission and exchange contributions is by studying decays other thanB 0 → Λ + cp , Σ + cp , which receive contributions from both A ex and A em . By taking A ex as the primary contribution, we predict that which are all within the capability of the current B factories.